Remove intersections between many objects
You may use a Direct Union of a BoolTool add on.


Activate the add on in User Preferences (Ctrl+Alt+U).


Select all the spheres with Shift+RMB, then press Spacebar and type direct union.


If you want to separate each object use Linked Flat Faces option.


Select one face of the sphere, go to Select-->Linked Flat Faces (or use a Shift+Ctrl+Alt+F shortcut). Increase the Sharpness value in a Tool Shelf. Then press P-->Selection to separete the selection.